Can a blue eyed mom and a brown eyed dad have a blue eyed baby?

Absolutely. The likelihood is not high, but the eye color combinations of father and mother can produce a blue-eyed child.

The theory behind this is called Mendelian genetics, pioneered by geneticist Gregor Mendel. He cross-bred different species of flowers to test the theory, and it has been refined over the years.

Sticking with the blue-eye vs. brown-eye topic, we know that brown eyes are the result of a dominant gene, and blue eyes are the result of a recessive gene. If we use "b" to represent the recessive gene for eye color, and we use "B" to represent the dominant gene, we can split the genes from the mother and father and combine them to come up with the relative probability of each child developing blue or brown eyes.

We know that a mother with blue eyes has two recessive genes, indicated "bb". A father, however, can have either one or two dominant genes, both will produce brown eyes. A father, therefore, can be indicated "BB" or "Bb".

Assuming a father of "BB" genes, and a mother of "bb" genes, all combinations of those genes will produce "Bb" genes in the children, producing only brown eyed children.

However, if one of those brown-eyed boys has a child with a blue-eyed mother, the combinations of genes will be 50% "Bb" and 50% "bb", so a 50% chance of a blue-eyed baby, and a 50% chance of a brown-eyed baby.

Mendelian genetics includes more factors than that, but in a simplistic way, this shows that there is the possibility that a brown-eyed dad and a blue-eyed mom can have a blue-eyed baby.

Coincidentally, I have brown eyes and my wife has blue eyes. My third son has blue eyes, indicating that I have at least one recessive gene in me.